Controllable intrinsic DC spin/valley Hall conductivity in ferromagnetic silicene: Exploring a fully spin/valley polarized transport
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Controllable intrinsic DC spin/valley Hallconductivity in ferromagnetic silicene:Exploring a fully spin/valley polarizedtransport
Yawar Mohammadi ∗ and Borhan Arghavani Nia Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran Department of Physics, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran
Abstract
We study intrinsic DC spin and valley Hall conductivity in doped ferromagnetic silicenein the presence of an electric filed applied perpendicular to silicene sheet. By calculatingits energy spectrum and wavefunction and by making use of Kubo formalism, we obtain ageneral relation for the transverse Hall conductivity which can be used to obtain spin- andvalley-Hall conductivity. Our results, in the zero limit of the exchange field, reduces tothe previous results. Furthermore we discuss electrically tunable spin and valley polarized ∗ Corresponding author. Tel./fax: +98 831 427 4569, Tel: +98 831 427 4569. E-mail address:[email protected] ransport in ferromagnetic silicene and obtain the necessary conditions for observing afully spin or valley polarized transport. Keywords: A. Ferromagnetic silicene; D. Tight-binding model; D: spin/valley Hall conduc-tivity; D: Fully spin/valley polarized transport.
Since successful isolation of a single layer of graphite[1], graphene, as the first real two-dimensional lattice structure which shows novel appealing properties[2, 3], many researchers tryto synthesis or isolate new two-dimensional materials. These efforts result in finding other twodimensional materials such as BN[4], transition metal dichalcogenides (TMDs)[5] and recentlya monolayer of silicon, known as silicene[6, 7, 8, 9].Silicene is a monolayer of silicon atoms arranged in a honeycomb lattice structure as similaras graphene. While, as in graphene, its low-energy dynamics near the two valleys at the corner ofthe hexagonal Brillouin zone is described by the Dirac theory, its Dirac electrons, due to a largespin-orbit (SO) interaction, are massive with a energy gap as 1 . meV [10, 11]. Furthermore,due to the large ionic radius, silicene is buckled[10] such that the A and B sublattices ofhoneycomb lattice shifted vertically with respect to each other and sit in two parallel planeswith a separation of 0 . nm [11, 12]. The buckled structure of silicene allows to tune its band gapvia an electric filed applied perpendicular to its layer. These features donate many attractiveproperties to silicene [11, 13, 14, 15, 16, 17, 18, 19, 20].The SO interaction in silicene is strong, so it is a suitable candidate to study the spin-basedeffects. Due to this fact, recently silicene has been the subject of strong interest[21, 22, 23, 24,25]. In addition to the spin degree of freedom which is the footstone of the spintronics, thevalley degree of freedom in silicene, as in graphene[26, 27, 28] and M oS [29, 30, 31, 32], can bemanipulated and hired in a new technology known as valleytronics. One can populate statespreferentially in one valley to achieve valley polarization. One way is to use circular polarizedlight which was discussed theoretically[29]. Another way is to apply a vertical external magneticfiled to silicene sheet, so Landau levels form in the electronic density of states. Then, if anexcitonic gap via an external vertical voltage included, n = 0 Landau level splits into distinctvalley- and spin-polarized levels[19]. This is in contrast to that occurs in graphene, in which n = 0 Landau level only splits between two distinct valley-polarized spin degenerated energylevels[33, 34, 35, 36]. In other way, as in graphene[37, 38], the spin/valley polarized current isobtained in silicene[39] junctions by deposing a ferromagnet on the top of its surface. Thesefeatures make silicene a promising candidate for spin- and valleytronic technology.In this paper, we consider DC valley/spin Hall conductivity in a ferromagnetic silicene (asilicene sheet with ferromagnet deposed on the top of its surface). We obtain a general relationfor its transverse Hall conductivity which can be use to calculate spin/valley Hall conductivityand to discuss possible phase transitions. Furthermore, we obtain the conditions necessary torealize fully valley/spin polarized transport, which depends on the doping, exchange magneti-zation and the applied perpendicular electric field. The paper is organized as follows. Sec.II isdevoted to introduce the Hamiltonian model and obtain the general relation for the transverseHall conductivity. In Sec.III we present our results and discussion. Finally in Sec.IV we endthis paper by summary and conclusions. 3 Model Hamiltonian
The low-energy dynamic in a ferromagnetic silicene, subjected to a uniform electric field appliedperpendicular to sislicene’s plane, is given by[10, 39] H η,s z = ¯ hv F ( k x τ x − ηk y τ y ) − ηs z ∆ so τ z + ∆ z τ z − s z M, (1)which acts in the sublattice pseudospin space with a wavefunction as Ψ η,s z = { ψ η,s z A , ψ η,s z B } T .The first part of the Hamiltonian is the Dirac hamiltonian describing the low-energy excitationsaround Dirac points ( K and K ′ denoted by η index) at the corners of the hexagonal first Brilouinzone. This term arises from nearest neighbor energy transfer. v F = is the Fermi velocity ofsilicene, τ i ( i = x, y, z ) are the Pauli matrixes and k = ( k x , k y ) is the two dimensional momentummeasured from Dirac points. The second term is the Kane-Mele term for the intrinsic spin-orbitcoupling, where ∆ so = 3 . meV [10] denotes to the spin-orbit coupling and s z index referred totwo spin degrees of freedom, up ( s z = +) and down ( s z = − ). The third term is the on-sitepotential difference between A and B sublattice, arising from the buckled structure of silicenewhen a perpendicular electric field is applied with ∆ z = E z d where E z is the electric field andthe 2 d = 0 . nm is the vertical separation of two different sublattice’s plane. The last term isthe exchange magnetization where M is the exchange field. The exchange field my be due toproximity effect arising from a magnetic adatom deposed on the surface of the silicene[40] orfrom a magnetic insulator substrate like EuO as proposed for graphene[37].We obtain the energy spectrum, by diagonalizing the Hamiltonian matrix given in Eq.(1),as ε η,s z ν = ν q ∆ η,s z + (¯ hv F k ) − s z M, (2)4here ν = +( ν = − ) denotes the conduction (valance) bands, ∆ η,s z = ηs z ∆ so − ∆ z and k = q k x + k y . The corresponding wavefunctions are given byΨ η,s z ν ( k ) = e i k . r √ χ η,s z q χ η,s z − ν ∆ η,s z ν q χ η,s z + ν ∆ η,s z e − iηφ k , (3)where χ η,s z = q ∆ η,s z + (¯ hv F k ) and φ k = tan − ( k y /k x ). Figure 1 shows the energy spectrum ofsilicene (Fig. 1(a)) and ferromagnetic silicene with M = ∆ so / z = 0 plotted in Fig. 1(b), ∆ z = ∆ so in Fig. 1(c) and ∆ z = 2∆ so in Fig. 1(d).These figures shows the energy spectrum around K . The energy spectrum around K ′ for zeroelectrical potential, ∆ z = 0, is equal to that of K point. To obtain the other energy-spectrumplots it is enough to reflect the energy-band plots with respect to E = 0 and exchange spin upand down.DC transverse Hall conductivity, σ xy , written in the Kubo formalism, is given by[41, 42] σ η,s z xy = − i e ¯ hA X k f ( ε η,s z + ) − f ( ε η,s z − )( ε η,s z + − ε η,s z − ) × h Ψ η,s z − | v y | Ψ η,s z + ih Ψ η,s z + | v x | Ψ η,s z − i , (4)where A is the area of the sample and velocity components can be obtained from the Hamil-tonian and using relation v k i = h ∂H∂k i . Furthermore f ( ε η,s z ν ) = 1 / (1 + e β ( ε η,szν − µ ) ) is Fermi-Diracdistribution function with µ being the chemical potential which at zero temperature is equalto Fermi energy. After calculating the expectation values of the velocities from Eq. 3 andsubstituting them in Eq. 4 we get σ η,s z xy = η e v F ¯ h π Z kdk ∆ η,s z χ η,s z ( f ( ε η,s z + ) − f ( ε η,s z − )) . (5)We restrict our consideration to zero temperature. So we can solve this equation analyticallyand obtain a general relation for σ η,s z xy for all arbitrary values of exchange field, M , and Fermi5nergy, µ which is σ η,s z xy = − η e π ¯ h sgn (∆ η,s z ) , (6)when | µ + s z M | < | ∆ η,s z | and σ η,s z xy = − η e π ¯ h ∆ η,s z | µ + s z M | , (7)when | µ + s z M | > | ∆ η,s z | . These equations are the main result of this paper. Here sgn(x) is signfunction which is 1 for x >
0, 0 for x = 0 and -1 for x <
0. The conditions | µ + s z M | < | ∆ η,s z | and | µ + s z M | > | ∆ η,s z | determine boundaries which separate different phase states. This can beexamined by calculating the spin- and valley-Hall conductivity which are defined[42, 43, 44, 45]as σ sxy = ¯ h e P η,s z s z σ η,s z xy and σ vxy = e P η,s z ησ η,s z xy respectively. This will be explained furtherin the next section where we present our results. In this section we present our results. First we examine our general result in the zero limitof M and µ . When M = 0 and µ = 0, only | µ + s z M | < | ∆ η,s z | is satisfied, so we have σ η,s z xy = − η e h sgn (∆ η,s z ). This yields σ sxy = − e π [ sgn (∆ + , + ) − sgn (∆ + , − )] , (8) σ vxy = − e π ¯ h [ sgn (∆ + , + ) + sgn (∆ + , − )] , (9)for DC spin- and valley-Hall conductivity of a silicene sheet, if an electric field applied per-pendicular to its plane. When 0 ≤ | ∆ z | < ∆ so these equations yield σ sxy = − e π and σ vxy = 0,6ndicating an intrinsic quantized spin Hall conductivity beside a zero valley Hall conductivity.This regime, as mentioned in previous works[14, 21], corresponds to a topological insulating(TI) phase characterized by a quantized nonzero spin-Hall conductivity which arises from thepresence of gapless helical edge mode. When | ∆ z | becomes equal to ∆ so , we have σ sxy = − e π and σ vxy = − e π ¯ h . In this regime silicene is a spin valley polarized metal (SVPM). If the electric filedincreases further such that ∆ so < | ∆ z | , the spin and valley Hall conductivity become σ sxy = 0and σ vxy = − e π ¯ h corresponding to a intrinsic quantum valley Hall effect. As mentioned[14], inthis regime silicene is a band insulator. These results indicate an electrically tunable phasetransition from a topological insulator to a spin valley polarized metal and then to a bandinsulator, as the electric field increases. It is evident that our results are in agreement with thesimilar results obtained for the DC spin- and valley-Hall conductivity of silicene in the previousworks[21, 22, 23].In the doped case, when the Fermi level locate inside the gap, obtained results for σ sxy and σ vxy are similar to those of the undoped case. Then, if an external vertical voltage is appliedand increases, a phase transition from a topological insulator to a metal and then to a bandinsulator occurs[21]. The boundary conditions which limit different phases are represented by − ∆ z − ∆ so < µ < ∆ z + ∆ so and ∆ z − ∆ so < µ < − ∆ z + ∆ so lines for topological insulatingphase and − ∆ z + ∆ so < µ < ∆ z − ∆ so or ∆ z + ∆ so < µ < − ∆ z − ∆ so lines for band insulatingphase. If the chemical potential increases further such that the Fermi level crosses the lowerconduction (upper valance) band, the results become σ sxy = − e π [ ∆ + , + | µ | + 1] , σ vxy = − e π ¯ h [ ∆ + , + | µ | −
1] (10)Notice that in this case, unlike the udoped case, silicene shows a nonzero DC response for both7pin- and valley-Hall conductivity which are controlled by ∆ so and ∆ z . While when the Fermilevel crosses both conduction (valance) bands, DC spin- and valley Hall conductivity become σ sxy = − e π ∆ so | µ | , σ vxy = e π ¯ h ∆ z | µ | . (11)On can see that in this case the DC spin-Hall conductivity is an intrinsic property and is onlycontrolled by ∆ so , whereas the valley-Hall conductivity is arising from and tuned by the appliedvertical voltage. Moreover, the direction of the valley Hall conductivity changes by inverting thedirection of the applied electric filed while the direction of the spin Hall conductivity remainsunchanged. These results for the doped case are in agreement with the results reported in theprevious works[21, 22, 23]. All these results have been summarized in Figs.2 and 3 where wehave shown the spin and valley Hall conductivity of a electron/hole doped silicene as a functionof the vertical electric filed and the Fermi energy.It is evident that in these cases, due to the symmetry of the band structure with respect tointerchanging the valley and spin index, there is no spin or valley polarization.Furthermore, the DC spin- and valley-Hall conductivity of a undoped ferromagnetic silicenecan be obtained by making use of Eqs. 6 and 7. When the exchange field is less than theminimum gap, at low external vertical voltage, silicene shows a quantum spin Hall effect withan intrinsic quantized Hall conductivity, σ sxy = − e π . By increasing the external vertical voltagesilicene becomes metal with nonzero charge- spin- and valley-Hall conductivity. Then at highexternal vertical voltages a phase transition to a conventional band insulator (characterized byzero charge and spin-Hall conductivities and a nonzero quantized valley-Hall conductivity, σ vxy = − e π ¯ h ) occurs. The necessary conditions to realize a quantum spin Hall effect are represented by − ∆ z − ∆ so < µ < ∆ z +∆ so and ∆ z − ∆ so < µ < − ∆ z +∆ so lines. While the similar conditions for8 quantum valley Hall effect are − ∆ z + ∆ so < M < ∆ z − ∆ so or ∆ z + ∆ so < M < − ∆ z − ∆ so .Furthermore, when the Fermi level crosses the lower conduction (upper valance) band, ourresults for the DC spin and valley Hall conductivity become σ sxy = − e π [ ∆ + , + | M | + 1] , σ vxy = − e π ¯ h [ ∆ + , + | M | −
1] (12)Then if the exchange filed increases further such that the Fermi level crosses both conduction(valance) bands, DC spin- and valley Hall conductivity become σ sxy = − e π ∆ so | M | , σ vxy = e π ¯ h ∆ z | M | . (13)Note, if the Fermi level crosses both conduction (valance) bands, the DC spin and valley Hallconductivity in a ferromagnetic silicene are intrinsic properties which are controlled only by ∆ so and ∆ z respectively. Moreover, one can see that in a undoped ferromagnetic silicene similarto the previous case, due to the symmetry of the band structure with respect to interchangingthe valley and spin index, any spin/valley polarized transport can not be attained. To achievea valley or spin polarized transport, one can populate spin or valley states differentially in onevalley or spin state. This can be attained, as explained in the following, in a doped ferromagneticsilicene.Our results for the spin- and valley Hall conductivity in a doped ferromagnetic silicene havebeen summarized in Fig.4 and Fig. 5. One can see that the region, in which a quantized spinor valley Hall effect occur, becomes limited when the induced exchange field increases. Thenecessary conditions to realize a quantum spin Hall effect are represented by − ∆ z − ∆ so < µ < ∆ z + ∆ so and ∆ z − ∆ so < µ < − ∆ z + ∆ so lines. While the similar conditions for a quantumvalley Hall effect are − ∆ z + ∆ so < M < ∆ z − ∆ so or ∆ z + ∆ so < M < − ∆ z − ∆ so .9oreover, as mentioned above, a partial or fully spin/valley polarized transport can becaptured in a doped ferromagnetic silicene. This can be seen in Fig.6 and Fig. 7. In Fig.6 wehave shown our results for σ ↑ xy = σ K ↑ xy + σ K ′ ↑ xy and σ ↓ xy = σ K ↓ xy + σ K ′ ↓ xy . The regions, in which σ ↑ xy ( e π ) or σ ↓ xy ( e π ) is zero, determine the necessary conditions to realize a fully spin polarizedtransport. These regions have been shown in Fig. 8. Furthermore, in Fig. 7 the plots of σ Kxy = σ K ↑ xy + σ K ↓ xy and σ K ′ xy = σ K ′ ↑ xy + σ K ′ ↓ xy , as a function of the vertical electric filed and thechemical potential, have been shown. The regions, in which a fully valley polarized transportcan be detected, have been shown in Fig. 8. In summary we studied the intrinsic DC valley and spin Hall conductivity in a ferromagneticsilicene, exploring a fully spin or valley polarized transport. First we calculated its eigenvaluesand eigenfunctions. Then, by making use of the Kubo formula, we derived a general relationfor the spin and valley Hall conductivity of the ferromagnetic silicene in the presence of finitedoping and an electric filed applied perpendicular to its plane. 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B 86, (2012) 205425-205432.14 F k x /2 π∆ so ε / ∆ s o −4 −2 0 2 4−4−2024 hv F k x /2 π∆ so ε / ∆ s o −4 −2 0 2 4−4−2024 hv F k x /2 π∆ so ε / ∆ s o −4 −2 0 2 4−4−2024 hv F k x /2 π∆ so ε / ∆ s o d b a c Figure 1: The energy spectrum of (a) silicene, and ferromagnetic silicene with M = ∆ so / z = 0, (c) ∆ z = ∆ so and (d) ∆ z = 2∆ so .15 z / ∆ so µ / ∆ s o −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2−1.5−1−0.500.511.52 −2−1.5−1−0.50 Figure 2: The transverse spin Hall conductivity, σ sxy ( e π ), in a doped silicene as a function ofthe vertical electric filed and the chemical potential.16 z / ∆ so µ / ∆ s o −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2−1.5−1−0.500.511.52 −2−1.5−1−0.500.511.52 Figure 3: The transverse valley Hall conductivity, σ vxy ( e π ¯ h ), in a doped silicene as a function ofthe vertical electric filed and the chemical potential.17 z / ∆ so µ / ∆ s o −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2−1.5−1−0.500.511.52 −2−1.5−1−0.50 Figure 4: The transverse spin Hall conductivity, σ sxy ( e π ), in a doped ferromagnetic silicene with M = ∆ so /
2, as a function of the vertical electric filed and the chemical potential.18 z / ∆ so µ / ∆ s o −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2−1.5−1−0.500.511.52 −2−1.5−1−0.500.511.52 Figure 5: The transverse valley Hall conductivity, σ vxy ( e π ¯ h ), in a doped ferromagnetic silicenewith M = ∆ so /
2, as a function of the vertical electric filed and the chemical potential.19 z / ∆ so µ / ∆ s o −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−202 ∆ z / ∆ so µ / ∆ s o −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−202 00.51−1−0.50 Figure 6: Plots of σ ↑ xy ( e π ) (top) and σ ↓ xy ( e π ) (bottom), in a doped ferromagnetic silicene with M = ∆ so /
2, as a function of the vertical electric filed and the chemical potential. The regionsin which σ ↑ xy ( e π ) or σ ↓ xy ( e π ) is zero determine the necessary conditions to realize a fully spinpolarized transport. 20 z / ∆ so µ / ∆ s o −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−202 ∆ z / ∆ so µ / ∆ s o −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−202 −101−101 Figure 7: Plots of σ Kxy ( e π ¯ h ) (top) and σ K ′ xy ( e π ¯ h ) (bottom), in a doped ferromagnetic silicene with M = ∆ so /